An outlier is something “abnormal” that “sticks out”. For example, the noise that “protrudes” from background noise. Such noise would typically be, in terms of its amplitude distribution, non-Gaussian. What is actually observed may depend on a source, the way noise couples into a system, and where in the system it is observed. Hence various particular instances of outlier noise may be known under different names, including, but not limited to, such as impulsive noise, transient noise, sparse noise, platform noise, burst noise, crackling noise, clicks & pops, and others. Depending on the way noise couples into a system and where in the system it is observed, noise with the same origin may have different appearances, and may or may not even be seen as an outlier noise.
Non-Gaussian (and, in particular, outlier) noise affecting communication and data acquisition systems may originate from a multitude of natural and technogenic (man-made) phenomena in a variety of applications. Examples of natural outlier (e.g. impulsive) noise sources include ice cracking (in polar regions) and snapping shrimp (in warmer waters) affecting underwater acoustics [1-3]. Electrical man-made noise is transmitted into a system through the galvanic (direct electrical contact), electrostatic coupling, electromagnetic induction, or RF waves. Examples of systems and services harmfully affected by technogenic noise include various sensor, communication, and navigation devices and services [4-15], wireless internet [16], coherent imaging systems such as synthetic aperture radar [17], cable, DSL, and power line communications [18-24], wireless sensor networks [25], and many others. An impulsive noise problem also arises when devices based on the ultra-wideband (UWB) technology interfere with narrowband communication systems such as WLAN [26] or CDMA-based cellular systems [27]. A particular example of non-Gaussian interference is electromagnetic interference (EMI), which is a widely recognized cause of reception problems in communications and navigation devices. The detrimental effects of EMI are broadly acknowledged in the industry and include reduced signal quality to the point of reception failure, increased bit errors which degrade the system and result in lower data rates and decreased reach, and the need to increase power output of the transmitter, which increases its interference with nearby receivers and reduces the battery life of a device.
A major and rapidly growing source of EMI in communication and navigation receivers is other transmitters that are relatively close in frequency and/or distance to the receivers. Multiple transmitters and receivers are increasingly combined in single devices, which produces mutual interference. A typical example is a smartphone equipped with cellular, WiFi, Bluetooth. and GPS receivers, or a mobile WiFi hotspot containing an HSDPA and/or LTE receiver and a WiFi transmitter operating concurrently in close physical proximity. Other typical sources of strong EMI are on-board digital circuits, clocks, buses, and switching power supplies. This physical proximity, combined with a wide range of possible transmit and receive powers, creates a variety of challenging interference scenarios. Existing empirical evidence [8, 28, 29] and its theoretical support [6, 7, 10] show that such interference often manifests itself as impulsive noise, which in some instances may dominate over the thermal noise [5, 8, 28].
A simplified explanation of non-Gaussian (and often impulsive) nature of a technogenic noise produced by digital electronics and communication systems may be as follows. An idealized discrete-level (digital) signal may be viewed as a linear combination of Heaviside unit step functions [30]. Since the derivative of the Heaviside unit step function is the Dirac δ-function [31], the derivative of an idealized digital signal is a linear combination of Dirac δ-functions, which is a limitlessly impulsive signal with zero interquartile range and infinite peakedness. The derivative of a “real” (i.e. no longer idealized) digital signal may thus be viewed as a convolution of a linear combination of Dirac δ-functions with a continuous kernel. If the kernel is sufficiently narrow (for example, the bandwidth is sufficiently large), the resulting signal would appear as an impulse train protruding from a continuous background signal. Thus impulsive interference occurs “naturally” in digital electronics as “di/dt” (inductive) noise or as the result of coupling (for example, capacitive) between various circuit components and traces, leading to the so-called “platform noise” [28]. Additional illustrative mechanisms of impulsive interference in digital communication systems may be found in [6-8, 10, 32].
The non-Gaussian noise described above affects the input (analog) signal. The current state-of-art approach to its mitigation is to convert the analog signal to digital, then apply digital nonlinear filters to remove this noise. There are two main problems with this approach. First, in the process of analog-to-digital conversion the signal bandwidth is reduced (and/or the ADC is saturated), and an initially impulsive broadband noise would appear less impulsive [7-10, 32]. Thus its removal by digital filters may be much harder to achieve. While this can be partially overcome by increasing the ADC resolution and the sampling rate (and thus the acquisition bandwidth) before applying digital nonlinear filtering, this further exacerbates the memory and the DSP intensity of numerical algorithms, making them unsuitable for real-time implementation and treatment of non-stationary noise. Thus, second, digital nonlinear filters may not be able to work in real time, as they are typically much more computationally intensive than linear filters. A better approach would be to filter impulsive noise from the analog input signal before the analog-to-digital converter (ADC), but such methodology is not widely known, even though the concepts of rank filtering of continuous signals are well understood [32-37].
Further, common limitations of nonlinear filters in comparison with linear filtering are that (1) nonlinear filters typically have various detrimental effects (e.g., instabilities and intermodulation distortions), and (2) linear filters are generally better than nonlinear in mitigating broadband Gaussian (e.g. thermal) noise.